A unified framework of demographic time

Demographic thought and practice is largely conditioned by the Lexis diagram, a two-dimensional graphical representation of the identity between age, period, and birth cohort. This relationship does not account for remaining years of life or other related time measures, whose use in demographic research is both underrepresented and incompletely situated. We describe a three-dimensional relationship between six different measures of demographic time: chronological age, time to death, lifespan, time of birth, time of death, and period. We describe four identities among subsets of these six measures, and a full identity that relates the six of them. One of these identities is the age-period-cohort identity, while the other three are relatively novel. We provide a topological overview of the diagrams that pertain to these identities. The 3-d geometric representation of the full six-way identity is proposed as a coordinate system that fully describes temporal variation in demographic data. We offer this framework as an instrument to enable the discovery of yet-undescribed relationships and patterns in formal and empirical demography.